In this exercise, use properties of rational exponents to simplify the given expression. Assume that all variables represent positive numbers.

$$\left(5^{\frac{2}{3}}\right)^3$$

Explain how to solve a radical equation with rational exponents.

Assume that all variables represent positive real numbers.

In this exercise, rationalize the given denominator. Simplify, if possible.

$$\frac{12}{\sqrt[3]{9}}$$

In this exercise, divide and simplify to the form $a + bi$.

$$\frac{2 i}{1+i}$$

In this exercise, assume that all variables represent positive real numbers.

In this exercises, simplify the radical expression and then rationalize the denominator.

$\frac{9}{\sqrt{3 x^2 y}}$

In this Exercise Set, assume that all variables represent positive real numbers.

In this exercise, divide and, if possible, simplify.

$$\frac{\sqrt[3]{x^2+7 x+12}}{\sqrt[3]{x+3}}$$

Explain why $\sqrt{x}=-1$ has no solution.

In this exercise, multiply and simplify. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers.

$$\sqrt{8 x} \cdot \sqrt{10 y}$$

In this exercise, use properties of rational exponents to simplify the given expression. Assume that all variables represent positive numbers.

$$\frac{x^{\frac{1}{4}}}{x^{\frac{3}{5}}}$$

In this exercise, assume that all variables represent positive real numbers.

In this exercises, rationalize the denominator.

$$\frac{3 x y^2}{\sqrt[5]{8 x y^3}}$$

In this exercise, find the indicated root, or state that the expression is not a real number.

$$-\sqrt[4]{81}$$

In this exercise, divide and simplify to the form $a + bi$.

$$\frac{3}{4+i}$$

Assume that all variables represent positive real numbers.

In this exercise, rationalize the given denominator. Simplify, if possible.

$$\sqrt{\frac{2}{7}}$$

In this exercise, multiply and simplify. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers.

$$\sqrt{5 x} \cdot \sqrt{10 y}$$